More on the world of Ur

A typical game of ROGOUR lasts between 177 and 243 turns (A turn is one roll of the dice followed by a legal move, if any). There is no longest game. With enough zeros you can get to any length. The shortest possible “cooperative” game is 42 turns: Green hops from home to d to 4 to 7, Red makes some meaningless move, Green bears off with a 3, another meaningless move by Red, a total of 6 turns, and the cycle repeats six more times for the remaining pieces.

This is kind of obvious when you think about it. Slightly more surprising is the fact that the shortest perfect is 42 turns as well. I wish Douglas Adams was alive and I could write and tell him about it. A perfect game is one played between two perfect players (two Ishtars), where each move is the best available. It looks slightly different than the “cooperative” game above since Red’s dice has to be carefully crafted so she has no opportunity to disrupt Green. Needless to say, it is a feast of fours for Green and threes and zeros for Red. You can find the game record in the games directory (shortestPerfectGame.ur).

The expected number of turns in a game is 209.810273. This is exact, and computed in a similar way as position probabilities, that is, by solving an enormous system of equations with 137,913,936 unknowns, using the same shortcuts and techniques we used before. This is for Ishtar vs. Ishtar games. The numbers are different when other players get involved. It falls to around 195 when Ishtar plays Santa (1820 ELO), and to 190 when she plays against Sam (1650 ELO). But it is 210 again for Santa vs. Sam, so it seems to depend on the difference in skill. The larger the difference, the shorter the game. Hardly surprising.

Due to the simple nature of the rules every board position is reachable from the starting position. You can always “go backwards”, that is find a piece to move backwards. What might be slightly more surprising is that every position is reachable in at most 56 turns. Those max positions are all of the “two-sided-bearoff” variety:

[0] ....  OO (5)     [0] ....  O. (6)     [0] ....  O. (6)
    ........             ........             ........
[0] ....  XX (5)     [0] ....  XX (5)     [0] ....  .X (6)

And so on.

Still, even though all positions are reachable, some might never appear with good play. That is, never reached in a game of two perfect players. Actually I expected this set to be quite large, but I was wrong. Only 5,941,240 positions (4.3%) never appear under perfect play. That means that a ROGOUR expert needs to be familiar with almost any type of position.